NCERT Class X Chapter 5: Arithmetic Progression Example 13

Question:

How many terms of the AP : 24, 21, 18, . . . must be taken so that their sum is 78?

Given:

An arithmetic progression (AP): 24, 21, 18, ... Sum of terms = 78

To Find:

The number of terms (n) to be taken so that their sum is 78.

Formula:

Sum of n terms of an AP: S_n = n 2 [2a + (n - 1)d]

where a = first term, d = common difference, n = number of terms

Solution:

Here, a = 24, d = 21 - 24 = -3, S_n = 78

Using the formula for the sum of an AP:

78 = n 2 [2(24) + (n - 1)(-3)]

⇒ 156 = n[48 - 3n + 3]

⇒ 156 = n[51 - 3n]

⇒ 156 = 51n - 3n²

⇒ 3n² - 51n + 156 = 0

Dividing by 3:

n² - 17n + 52 = 0

⇒ (n - 4)(n - 13) = 0

⇒ n = 4 or n = 13

Result:

The number of terms required is either 4 or 13.

Next question solution:

NCERT Class X Chapter 5: Arithmetic Progression Example 14-i

Explore more in Arithmetic Progressions chapter:

Click this link to explore more NCERT Class X Chapter 5 Arithmetic Progressions solutions

Explore more:

Click this link to explore more NCERT Class X chapter solutions

© Kaliyuga Ekalavya. All rights reserved.